Counting subgroups of the groups ${\Bbb Z}_{n_1} \times \cdots \times {\Bbb Z}_{n_k}$: a survey

TL;DR

This survey reviews existing exact and asymptotic formulas for counting cyclic subgroups and total subgroups in direct products of finite cyclic groups, providing a comprehensive overview of the topic.

Contribution

It compiles and discusses known formulas and asymptotic results for subgroup enumeration in these groups, highlighting recent advances and open problems.

Findings

Summarizes exact formulas for subgroup counts.

Provides asymptotic estimates for large parameters.

Identifies gaps and future directions in subgroup enumeration.

Abstract

We present a survey of exact and asymptotic formulas on the number of cyclic subgroups and total number of subgroups of the groups ${\Bbb Z}_{n_1} \times \cdots \times {\Bbb Z}_{n_k}$, where $k\ge 2$ and $n_1,\ldots,n_k$ are arbitrary positive integers.